Probability distribution Bernoulli, binomial, Beta

Bernoulli, binomial, Beta distribution detailed

This paper focuses on the distribution of random variable associated with discrete stochastic variables, discrete random variables continuous random variable such as classic Gaussian distributions (Gaussian distribution) will be in other article mediations Shaoxing.

What exactly is distribution "distribution".
Distribution is an abstract concept, from the perspective of probability statistics, distribution is a mathematical function that providing the probabilities of occurrence of different possible outcomes in an Experim Ent.
In layman's terms, distribution can be seen as a description of random variable stochastic variables, and distributions can give the probability of the value of all random variables (continuous random variables are the probability pdf), which is the forward process.
The results we really observe are just samples, and the results we predict are just expectations, and for the distribution of random variables, we can only guess by a large number of samples (and also with a priori knowledge), which is the reverse process.

Typical discrete random variable distributions are: Bernoulli distribution (Bernoulli distribution) Two-item distribution (binomial distribution) classification distribution (categorical distribution) Multi-item distribution (multinomial distribution)

And their conjugate prior distribution: Beta distribution (beta distribution) Dirichlet distribution (Dirichelet distribution)

Seemingly a lot of distributions, but are based on simple Bernoulli, very regular, and many properties are the same. This paper will introduce the distribution from Bernoulli, give the maximum likelihood deduction application, and finally introduce the concept of conjugacy. 1. Detailed distribution 1.1 Bernoulli distribution (also known as 01 distribution)

This is the most basic discrete random variable probability distribution, equivalent to the statistical theory of Hello world, we remember a variable y∈{1, 0} y\in \{1,\ 0\}, that an event or experiment has only two results, 1 or 0, can be understood to occur or not occur; For example, the classic coin toss experiment, We can only get two results, heads or tails.

We usually introduce μ∈[0,1] \mu \in [0, 1] to indicate the probability of getting the result Y=1 Y=1 (for example, the coin faces up):
P (y=1|μ) =μ, y∈{1, 0} p (y=1|\mu) =\mu,\ y\in \{1,\ 0\}
This type of distribution is called the Bernoulli distribution:
Bern (y|μ) =μy (1−μ) 1−y Bern (Y|\MU) =\mu^y (1-\MU) ^{1-y}
There is no simpler distribution than this, only one parameter is defined, and there are two possible outcomes. But Bern in machine learning is extremely common, for example, the value of a Logistic function is actually an estimate of the μ\mu of parameters in the Bernoulli distribution.

1.2 Bernoulli distribution and two distribution, classification distribution, multi-item distribution

In general, the Bernoulli distribution is the basis, the binomial distribution is extended to n experiments, the classification distribution is extended to a single test k results, the distribution of multiple distributions to n trials and each k results. Bernoulli: A single randomized trial with only two possible outcomes; can also be called a two-item distribution at n=1. Bern (y|μ) =μy (1−μ) 1−y Bern (Y|\MU) =\mu^y (1-\MU) ^{1-y} binomial: n independent Bernoulli test to obtain the discrete distribution of the results of a successful number of times; For example, a discrete probability distribution of 5 toss coins, with a number of heads facing up (there are 6 possible). Bin (m| n,μ) =n!m! (n−m)!μm (1−μ) (n−m) Bin (m| N,\MU) =\frac{n!} {m! (N-M)!} \mu^m (1-\MU) ^{(n-m)} categorical: promotion of Bernoulli, also a single randomized trial with K possible results (mutual exclusion) , rarely useful, formulas and multinomial Similar (remove factorial part), do not write. Multinomial: n Independent tests, each trial has k possible results (mutual exclusion) , each test is also known as categorical, can be said to be the most general distribution. The most common example is throwing the dice. Mult (m1,m2,...  ,mk|μ1,μ2,...  ,μ<